{
  "id": "2007.09873",
  "version": "v1",
  "published": "2020-07-20T03:53:53.000Z",
  "updated": "2020-07-20T03:53:53.000Z",
  "title": "A Birkhoff-Bruhat Atlas for partial flag varieties",
  "authors": [
    "Huanchen Bao",
    "Xuhua He"
  ],
  "comment": "22 pages, 1 figure",
  "categories": [
    "math.RT",
    "math.AG",
    "math.CO"
  ],
  "abstract": "A partial flag variety ${\\mathcal {P}}_K$ of a Kac-Moody group $G$ has a natural stratification into projected Richardson varieties. When $G$ is a connected reductive group, a Bruhat atlas for ${\\mathcal {P}}_K$ was constructed by He, Knutson and Lu: ${\\mathcal {P}}_K$ is locally modeled with Schubert varieties in some Kac-Moody flag variety as stratified spaces. The existence of Bruaht atlases implies some nice combinatorial and geometric properties on the partial flag varieties and the decomposition into projected Richardson varieties. A Bruhat atlas does not exist for partial flag varieties of an arbitrary Kac-Moody group due to combinatorial and geometric reasons. To overcome obstructions, we introduce the notion of Birkhoff-Bruhat atlas. Instead of the Schubert varieties used in a Bruhat atlas, we use the $J$-Schubert varieties for a Birkhoff-Bruhat atlas. The notion of the $J$-Schubert varieties interpolates Birkhoff decomposition and Bruhat decomposition of the full flag variety (of a larger Kac-Moody group). The main result of this paper is the construction of a Birkhoff-Bruhat atlas for any partial flag variety ${\\mathcal {P}}_K$ of a Kac-Moody group. We also construct a combinatorial atlas for the index set $Q_K$ of the projected Richardson varieties in ${\\mathcal {P}}_K$. As a consequence, we show that $Q_K$ has some nice combinatorial properties. This gives a new proof and generalizes the work of Williams in the case where the group $G$ is a connected reductive group.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-20T03:53:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14M15",
      "20F55",
      "20G44"
    ],
    "keywords": [
      "partial flag variety",
      "birkhoff-bruhat atlas",
      "kac-moody group",
      "projected richardson varieties",
      "schubert varieties interpolates birkhoff decomposition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}